Chapter 17 The Ideal Rectifier

Chapter 17 The Ideal Rectifier 17.1 Properties of the ideal rectifier 17.2 Realization of a near-ideal rectifier 17.3 Single-phase converter systems employing ideal rectifiers 17.4 RMS values of rectifier waveforms 17.5 Ideal three-phase rectifiers Fundamentals of Power Electronics 1 Chapter 17: The Ideal Rectifier

17.1 Properties of the ideal rectifier It is desired that the rectifier present a resistive load to the ac power system. This leads to unity power factor ac line current has same waveshape as voltage i ac = v ac R e i ac R e is called the emulated resistance v ac R e Fundamentals of Power Electronics 2 Chapter 17: The Ideal Rectifier

Control of power throughput P av = V 2 ac,rms R e (v control ) i ac Power apparently consumed by R e is actually transferred to rectifier dc output port. To control the amount of output power, it must be possible to adjust the value of R e. v ac R e (v control ) v control Fundamentals of Power Electronics 3 Chapter 17: The Ideal Rectifier

Output port model The ideal rectifier is lossless and contains no internal energy storage. Hence, the instantaneous input power equals the instantaneous output power. Since the instantaneous power is independent of the dc load characteristics, the output port obeys a power source characteristic. v ac i ac ac input p= R e (v control ) v control v 2 ac R e (v control ) Ideal rectifier (LFR) p = v ac 2 /R e i v dc output vi=p= v ac 2 R e Fundamentals of Power Electronics 4 Chapter 17: The Ideal Rectifier

The dependent power source i i i vi = p p v v p v power source power sink i-v characteristic Fundamentals of Power Electronics 5 Chapter 17: The Ideal Rectifier

Equations of the ideal rectifier / LFR Defining equations of the ideal rectifier: i ac = vi=p v ac R e (v control ) When connected to a resistive load of value R, the input and output rms voltages and currents are related as follows: V rms V ac,rms = R Re p= v 2 ac R e (v control ) I ac,rms I rms = R Re Fundamentals of Power Electronics 6 Chapter 17: The Ideal Rectifier

17.2 Realization of a near-ideal rectifier Control the duty cycle of a dc-dc converter, such that the input current is proportional to the input voltage: i g i ac dcdc converter 1 : M(d) i v ac v g v C R d i g Controller v g Fundamentals of Power Electronics 7 Chapter 17: The Ideal Rectifier

Waveforms v ac V M i g t v V i ac V M /R e t M M min v g V M v ac =V M sin (ωt) v g =V M sin (ωt) M(d) = v v g = M min = V V M V M V sin (ωt) Fundamentals of Power Electronics 8 Chapter 17: The Ideal Rectifier

Choice of converter M(d) = v v g = V M V sin (ωt) M M min To avoid distortion near line voltage zero crossings, converter should be capable of producing M(d) approaching infinity Above expression neglects converter dynamics Boost, buck-boost, Cuk, SEPIC, and other converters with similar conversion ratios are suitable We will see that the boost converter exhibits lowest transistor stresses. For this reason, it is most often chosen Fundamentals of Power Electronics 9 Chapter 17: The Ideal Rectifier

Boost converter with controller to cause input current to follow input voltage i g i ac Boost converter L D 1 i v ac v g Q 1 C v R v control Multiplier X v g R s i g v a PWM v ref = k x v g v control v err G c (s) Compensator Controller Fundamentals of Power Electronics 10 Chapter 17: The Ideal Rectifier

Variation of duty cycle in boost rectifier M(d) = v v g = V M V sin (ωt) Since M 1 in the boost converter, it is required that V V M If the converter operates in CCM, then M(d) = 1 1d The duty ratio should therefore follow d=1 v g V in CCM Fundamentals of Power Electronics 11 Chapter 17: The Ideal Rectifier

CCM/DCM boundary, boost rectifier Inductor current ripple is i g = v gdt s 2L Low-frequency (average) component of inductor current waveform is i g Ts = v g R e The converter operates in CCM when i g Ts > i g d< 2L R e T s Substitute CCM expression for d: R e < T s 2L 1 v g V for CCM Fundamentals of Power Electronics 12 Chapter 17: The Ideal Rectifier

CCM/DCM boundary R e < T s 2L 1 v g V for CCM Note that v g varies with time, between 0 and V M. Hence, this equation may be satisfied at some points on the ac line cycle, and not at others. The converter always operates in CCM provided that R e < 2L T s The converter always operates in DCM provided that R e > T s 2L 1 V M V For R e between these limits, the converter operates in DCM when v g is near zero, and in CCM when v g approaches V M. Fundamentals of Power Electronics 13 Chapter 17: The Ideal Rectifier

Static input characteristics of the boost converter v g A plot of input current i g vs input voltage v g, for various duty cycles d. In CCM, the boost converter equilibrium equation is v g V =1d The input characteristic in DCM is found by solution of the averaged DCM model (Fig. 10.12(b)): i g R e (d) p R Beware! This DCM R e (d) from Chapter 10 is not the same as the rectifier emulated resistance R e = v g /i g V Solve for input current: i g = v g R e (d) p V v g with p= v g 2 R e (d) R e (d) = 2L d 2 T s Fundamentals of Power Electronics 14 Chapter 17: The Ideal Rectifier

Static input characteristics of the boost converter Now simplify DCM current expression, to obtain 2L i VT g 1 v g s V = d 2 v g V CCM/DCM mode boundary, in terms of v g and i g : 2L i VT g > v g s V 1 v g V Fundamentals of Power Electronics 15 Chapter 17: The Ideal Rectifier

Boost input characteristics with superimposed resistive characteristic 1 0.75 d = 1 d = 0.8 d = 0.6 d = 0.4 d = 0.2 d = 0 CCM: v g V =1d j g = VT 2L i g s 0.5 0.25 i g =v g /R e DCM CCM DCM: 2L i VT g 1 v g s V CCM when 2L i VT g > v g s V = d 2 v g V 1 v g V 0 0 0.25 0.5 0.75 1 m g = v g V Fundamentals of Power Electronics 16 Chapter 17: The Ideal Rectifier

R e of the multiplying (average current) controller Solve circuit to find R e : Current sensor gain v ac i g i ac v control Multiplier v g v g R s i g Controller Boost converter L X v a v ref = k x v g v control v err Q 1 D 1 PWM G c (s) Compensator i C v R v a =i g R s when the error signal is small, v a v ref multiplier equation v ref =k x v g v control then R e is R e = v g i g = simplify: v ref k x v control v a R s R e (v control ) = R s k x v control Fundamentals of Power Electronics 17 Chapter 17: The Ideal Rectifier

Low frequency system model i g Ts Ideal rectifier (LFR) i Ts i ac p Ts v ac v g Ts R e C v Ts R R e R e = R s k x v control v control R e (v control ) = R s k x v control This model also applies to other converters that are controlled in the same manner, including buck-boost, Cuk, and SEPIC. Fundamentals of Power Electronics 18 Chapter 17: The Ideal Rectifier

Open-loop DCM approach We found in Chapter 10 that the buck-boost, SEPIC, and Cuk converters, when operated open-loop in DCM, inherently behave as loss-free resistors. This suggests that they could also be used as near-ideal rectifiers, without need for a multiplying controller. Advantage: simple control Disadvantages: higher peak currents, larger input current EMI Like other DCM applications, this approach is usually restricted to low power (< 200W). The boost converter can also be operated in DCM as a low harmonic rectifier. Input characteristic is i g Ts = v g v g2 R e vv g R e Input current contains harmonics. If v is sufficiently greater than v g, then harmonics are small. Fundamentals of Power Electronics 19 Chapter 17: The Ideal Rectifier

17.3 Single-phase converter systems containing ideal rectifiers It is usually desired that the output voltage v be regulated with high accuracy, using a wide-bandwidth feedback loop For a given constant load characteristic, the instantaneous load current and power are then also constant: p load =vi=vi The instantaneous input power of a single-phase ideal rectifier is not constant: p ac =v g i g with v g =V M sin (ωt) i g = v g R e so p ac = V 2 M sin R 2 ωt = V 2 M e 2R e 1 cos 2ωt Fundamentals of Power Electronics 20 Chapter 17: The Ideal Rectifier

Power flow in single-phase ideal rectifier system Ideal rectifier is lossless, and contains no internal energy storage. Hence instantaneous input and output powers must be equal An energy storage element must be added Capacitor energy storage: instantaneous power flowing into capacitor is equal to difference between input and output powers: p C = de C dt = d 1 2 Cv 2 C dt = p ac p load Energy storage capacitor voltage must be allowed to vary, in accordance with this equation Fundamentals of Power Electronics 21 Chapter 17: The Ideal Rectifier

Capacitor energy storage in 1 system p ac P load v c = d 1 2 Cv 2 C dt = p ac p load Fundamentals of Power Electronics 22 Chapter 17: The Ideal Rectifier t

Single-phase system with internal energy storage i g Ideal rectifier (LFR) i 2 p load = VI = P load v ac i ac v g R e p ac Ts C v C Dcdc converter v i load Energy storage capacitor Energy storage capacitor voltage v C must be independent of input and output voltage waveforms, so that it can vary according to = d 1 2 Cv 2 C dt = p ac p load This system is capable of Wide-bandwidth control of output voltage Wide-bandwidth control of input current waveform Internal independent energy storage Fundamentals of Power Electronics 23 Chapter 17: The Ideal Rectifier

Hold up time Internal energy storage allows the system to function in other situations where the instantaneous input and output powers differ. A common example: continue to supply load power in spite of failure of ac line for short periods of time. Hold up time: the duration which the dc output voltage v remains regulated after v ac has become zero A typical hold-up time requirement: supply load for one complete missing ac line cycle, or 20msec in a 50Hz system During the hold-up time, the load power is supplied entirely by the energy storage capacitor Fundamentals of Power Electronics 24 Chapter 17: The Ideal Rectifier

Energy storage element Instead of a capacitor, and inductor or higher-order LC network could store the necessary energy. But, inductors are not good energy-storage elements Example 100V 100µF capacitor 100A 100µH inductor each store 1 Joule of energy But the capacitor is considerably smaller, lighter, and less expensive So a single big capacitor is the best solution Fundamentals of Power Electronics 25 Chapter 17: The Ideal Rectifier

Inrush current A problem caused by the large energy storage capacitor: the large inrush current observed during system startup, necessary to charge the capacitor to its equilibrium value. Boost converter is not capable of controlling this inrush current. Even with d = 0, a large current flows through the boost converter diode to the capacitor, as long as v < v g. Additional circuitry is needed to limit the magnitude of this inrush current. Converters having buck-boost characteristics are capable of controlling the inrush current. Unfortunately, these converters exhibit higher transistor stresses. Fundamentals of Power Electronics 26 Chapter 17: The Ideal Rectifier

Universal input The capability to operate from the ac line voltages and frequencies found everywhere in the world: 50Hz and 60Hz Nominal rms line voltages of 100V to 260V: 100V, 110V, 115V, 120V, 132V, 200V, 220V, 230V, 240V, 260V Regardless of the input voltage and frequency, the near-ideal rectifier produces a constant nominal dc output voltage. With a boost converter, this voltage is 380 or 400V. Fundamentals of Power Electronics 27 Chapter 17: The Ideal Rectifier

Low-frequency model of dc-dc converter Dc-dc converter produces well-regulated dc load voltage V. Load therefore draws constant current I. Load power is therefore the constant value P load = VI. To the extent that dc-dc converter losses can be neglected, then dc-dc converter input power is P load, regardless of capacitor voltage v c. Dc-dc converter input port behaves as a power sink. A low frequency converter model is i 2 p load = VI = P load i C v C P load V v load Energy storage capacitor Dc-dc converter Fundamentals of Power Electronics 28 Chapter 17: The Ideal Rectifier

Low-frequency energy storage process, 1 system A complete low-frequency system model: i g i 2 i ac p ac Ts p load = VI = P load i v ac v g R e C v C P load V v load Ideal rectifier (LFR) Energy storage capacitor Dc-dc converter Difference between rectifier output power and dc-dc converter input power flows into capacitor In equilibrium, average rectifier and load powers must be equal But the system contains no mechanism to accomplish this An additional feeback loop is necessary, to adjust R e such that the rectifier average power is equal to the load power Fundamentals of Power Electronics 29 Chapter 17: The Ideal Rectifier

Obtaining average power balance i g i 2 p load = VI = P load i ac p ac Ts i v ac v g R e C v C P load V v load Ideal rectifier (LFR) Energy storage capacitor Dc-dc converter If the load power exceeds the average rectifier power, then there is a net discharge in capacitor energy and voltage over one ac line cycle. There is a net increase in capacitor charge when the reverse is true. This suggests that rectifier and load powers can be balanced by regulating the energy storage capacitor voltage. Fundamentals of Power Electronics 30 Chapter 17: The Ideal Rectifier

A complete 1 system containing three feedback loops v ac i g i ac v g Boost converter L D 1 Q 1 i 2 v C C DCDC Converter i Load v v control Multiplier X v g R s i g v a PWM d v ref1 = k x v g v control v err G c (s) Compensator Wide-bandwidth input current controller v Compensator and modulator v ref3 Wide-bandwidth output voltage controller v C Compensator v ref2 Low-bandwidth energy-storage capacitor voltage controller Fundamentals of Power Electronics 31 Chapter 17: The Ideal Rectifier

Bandwidth of capacitor voltage loop The energy-storage-capacitor voltage feedback loop causes the dc component of v c to be equal to some reference value Average rectifier power is controlled by variation of R e. R e must not vary too quickly; otherwise, ac line current harmonics are generated Extreme limit: loop has infinite bandwidth, and v c is perfectly regulated to be equal to a constant reference value Energy storage capacitor voltage then does not change, and this capacitor does not store or release energy Instantaneous load and ac line powers are then equal Input current becomes i ac = p ac v ac = p load v ac = P load V M sin ωt Fundamentals of Power Electronics 32 Chapter 17: The Ideal Rectifier

Input current waveform, extreme limit i ac = p ac v ac = p load v ac = P load V M sin ωt THD Power factor 0 v ac i ac t So bandwidth of capacitor voltage loop must be limited, and THD increases rapidly with increasing bandwidth Fundamentals of Power Electronics 33 Chapter 17: The Ideal Rectifier

17.4 RMS values of rectifier waveforms Doubly-modulated transistor current waveform, boost rectifier: i Q Computation of rms value of this waveform is complex and tedious Approximate here using double integral Generate tables of component rms and average currents for various rectifier converter topologies, and compare t Fundamentals of Power Electronics 34 Chapter 17: The Ideal Rectifier

RMS transistor current RMS transistor current is i Q I Qrms = 1 Tac 0 T ac i Q 2 dt Express as sum of integrals over all switching periods contained in one ac line period: t T ac /T s I Qrms = 1 T 1 s i 2 Q dt Tac Ts n =1 nt s (n-1)t s Quantity in parentheses is the value of i Q2, averaged over the n th switching period. Fundamentals of Power Electronics 35 Chapter 17: The Ideal Rectifier

Approximation of RMS expression T ac /T s n =1 I Qrms = 1 T 1 s i 2 Q dt Tac Ts nt s (n-1)t s When T s << T ac, then the summation can be approximated by an integral, which leads to the double-average: I Qrms 1 Tac T ac /T s nt s lim Ts T 1 0 s i 2 n=1 Q (τ)dτ Ts (n-1)t s = 1 1 i 2 Q (τ)dτdt Tac Ts 0 T ac t tt s = i Q 2 Ts T ac Fundamentals of Power Electronics 36 Chapter 17: The Ideal Rectifier

17.4.1 Boost rectifier example For the boost converter, the transistor current i Q is equal to the input current when the transistor conducts, and is zero when the transistor is off. The average over one switching period of i Q2 is therefore If the input voltage is i Q 2 tt s t = 1 i T s T 2 Q dt s = di 2 ac v ac =V M sin ωt then the input current will be given by i ac = V M Re and the duty cycle will ideally be sin ωt V v ac = 1 1d (this neglects converter dynamics) Fundamentals of Power Electronics 37 Chapter 17: The Ideal Rectifier

Boost rectifier example Duty cycle is therefore d=1 V M V Evaluate the first integral: 2 i Q = V 2 M T 2 s R e sin ωt Now plug this into the RMS formula: I Qrms = 1 Tac i Q 2 1 V M V sin ωt sin 2 ωt 0 T ac T s dt = 1 Tac 0 T ac 2 V M 2 R e 1 V M V sin ωt sin 2 ωt dt I Qrms = 2 2 V M Tac 2 sin 2 ωt V M sin 3 R e V 0 T ac /2 ωt dt Fundamentals of Power Electronics 38 Chapter 17: The Ideal Rectifier

Integration of powers of sin θ over complete half-cycle n 1 π 0 π sin n (θ)dθ 1 π 0 π sin n (θ)dθ = 2 2 4 6 (n 1) π 1 3 5 n 1 3 5 (n 1) 2 4 6 n if n is odd if n is even 1 2 π 2 1 2 3 4 3π 4 3 8 5 6 16 15π 15 48 Fundamentals of Power Electronics 39 Chapter 17: The Ideal Rectifier

Boost example: Transistor RMS current I Qrms = V M 2R e 1 8 3π V M V = I ac rms 1 8 3π V M V Transistor RMS current is minimized by choosing V as small as possible: V = V M. This leads to I Qrms = 0.39I ac rms When the dc output voltage is not too much greater than the peak ac input voltage, the boost rectifier exhibits very low transistor current. Efficiency of the boost rectifier is then quite high, and 95% is typical in a 1kW application. Fundamentals of Power Electronics 40 Chapter 17: The Ideal Rectifier

Table of rectifier current stresses for various topologies Table 17.2 Summary of rectifier current stresses for several converter topologies rms A verage Peak CCM boost Transistor I ac rms 1 8 3π V M V I ac rms 2 2 π 1 π 8 V M I ac rms 2 V Diode I dc 16 3π V V M I dc 2 I dc V VM Inductor I ac rms I ac rms 2 2 π I ac rms 2 CCM flyback, with n:1 isolation transformer and input filter Transistor, xfmr primary L 1 I ac rms 1 8 3π I ac rms V M nv I ac rms 2 2 π I ac rms 2 1 V n I ac rms 2 2 π I ac rms 2 C 1 I 8 V M ac rms 3π nv 0 I ac rms 2 max 1, V M nv Diode, xfmr secondary I dc 3 2 16 3π nv V M I dc 2I dc 1 nv V M Fundamentals of Power Electronics 41 Chapter 17: The Ideal Rectifier

Table of rectifier current stresses continued CCM SEPIC, nonisolated Transistor L 1 C 1 I ac rms 1 8 3π I ac rms V M I 2 2 ac rms V π I ac rms 2 1 V M V I ac rms 2 2 π I ac rms 8 3π V M V 0 I ac rms 2 I ac rms max 1, V M V L 2 Diode I ac rms V M V 3 2 I ac rms 2 I dc 3 2 16 3π V V M I dc V M V I ac rms V M V 2 2I dc 1 V V M CCM SEPIC, with n:1 isolation transformer transistor L 1 C 1, xfmr primary Diode, xfmr secondary I ac rms 1 8 3π I ac rms I 8 V M ac rms 3π nv V M nv I ac rms 2 2 π I ac rms 2 1 V M nv I ac rms 2 2 π I dc 3 2 16 3π nv V M I dc I with, in all cases, ac rms = I dc 2 V V M, ac input voltage = V M sin(ω t) dc output voltage = V I ac rms 2 Fundamentals of Power Electronics 42 Chapter 17: The Ideal Rectifier 0 I ac rms 2I dc 2 max 1, n 1 nv V M

Comparison of rectifier topologies Boost converter Lowest transistor rms current, highest efficiency Isolated topologies are possible, with higher transistor stress No limiting of inrush current Output voltage must be greater than peak input voltage Buck-boost, SEPIC, and Cuk converters Higher transistor rms current, lower efficiency Isolated topologies are possible, without increased transistor stress Inrush current limiting is possible Output voltage can be greater than or less than peak input voltage Fundamentals of Power Electronics 43 Chapter 17: The Ideal Rectifier

Comparison of rectifier topologies 1kW, 240Vrms example. Output voltage: 380Vdc. Input current: 4.2Arms Converter Transistor rms current Transistor voltage Diode rms current Transistor rms current, 120V Diode rms current, 120V Boost 2 A 380 V 3.6 A 6.6 A 5.1 A Nonisolated SEPIC Isolated SEPIC 5.5 A 719 V 4.85 A 9.8 A 6.1 A 5.5 A 719 V 36.4 A 11.4 A 42.5 A Isolated SEPIC example has 4:1 turns ratio, with 42V 23.8A dc load Fundamentals of Power Electronics 44 Chapter 17: The Ideal Rectifier

17.5 Ideal three-phase rectifiers Ideal 3ø rectifier, modeled as three 1ø ideal rectifiers: 3øac input dc output ø a i a R e p a ø b i b R e p b R v ø c i c R e p c Fundamentals of Power Electronics 45 Chapter 17: The Ideal Rectifier

Ideal 3 rectifier model Combine parallel-connected power sources into a single source p tot : 3øac input dc output ø a i a R e ø b i b R e p tot = p a p b p c R v ø c i c R e Fundamentals of Power Electronics 46 Chapter 17: The Ideal Rectifier

Value of p tot Ac input voltages: v an =V M sin ωt v bn =V M sin ωt 120 v cn =V M sin ωt 240 Instantaneous phase powers: 3øac input ø a ø b ø c i a i b i c R e R e R e p a p b p c dc output R v p a = v 2 an R e R e = V 2 M 2R e 1 cos 2ωt p b = v 2 bn = V 2 M 1 cos 2ωt 240 2R e p c = v cn 2 R e = V 2 M 1 cos 2ωt 120 2R e Total 3ø instantaneous power: p tot =p a p b p c = 3 2 V M 2 R e 2 nd harmonic terms add to zero total 3ø power p tot is constant Fundamentals of Power Electronics 47 Chapter 17: The Ideal Rectifier

Instantaneous power in ideal 3 rectifier p tot =p a p b p c = 3 2 V M 2 R e In a balanced system, the ideal 3ø rectifier supplies constant power to its dc output a constant power load can be supplied, without need for lowfrequency internal energy storage 3øac input ø a ø b ø c i a i b i c R e R e R e p tot = p a p b p c dc output R v Fundamentals of Power Electronics 48 Chapter 17: The Ideal Rectifier

17.5.1 Three-phase rectifiers operating in CCM 3øac input ø a ø b 3øacdc boost rectifier i a i b L L v 12 2 1 Q 1 i 1 i 2 i 3 D 1 Q 2 D 2 Q 3 D 3 C dc output Load v ø c i c L 3 v 20 v 10 0 Q 4 Q 5 Q 6 D 4 D 5 D 6 Uses six current-bidirectional switches Operation of each individual phase is similar to the 1ø boost rectifier Fundamentals of Power Electronics 49 Chapter 17: The Ideal Rectifier

The 3 acðdc boost rectifier Voltage-source inverter, operated backwards as a rectifier Converter is capable of bidirectional power flow Dc output voltage V must be greater than peak ac line-line voltage V L,pk. Ac input currents are nonpulsating. In CCM, input EMI filtering is relatively easy Very low RMS transistor currents and conduction loss The leading candidate to replace uncontrolled 3ø rectifiers Requires six active devices Cannot regulate output voltage down to zero: no current limiting cannot replace traditional buck-type controlled rectifiers Fundamentals of Power Electronics 50 Chapter 17: The Ideal Rectifier

Control of switches in CCM 3 ac-dc boost rectifier Pulse-width modulation: v 10 v Drive lower transistors (Q 4 Q 6 ) with complements of duty cycles of respective upper transistors (Q 1 Q 3 ). Each phase operates independently, with its own duty cycle. v 20 v 10 Ts = d 1 v Ts 0 0 d 1 T s T s t v v 20 Ts = d 2 v Ts 1 v 12 2 v 20 3 v 10 0 Q 1 Q 4 i 1 i 2 i 3 Q 2 Q D 3 1 D 2 D 3 Q 5 Q 6 D 4 D 5 D 6 0 d 2 T s T s t Fundamentals of Power Electronics 51 Chapter 17: The Ideal Rectifier v 30 v 0 v 30 Ts = d 3 v Ts 0 d 3 T s T s t Conducting Q devices: 1 / D 1 Q 4 / D 4 Q 2 / D 2 Q 5 / D 5 Q 3 / D 3 Q 6 / D 6 0

Average switch waveforms Average the switch voltages: v 10 v v 10 Ts = d 1 v Ts v 10 Ts = d 1 v Ts v 20 Ts = d 2 v Ts v 30 Ts = d 3 v Ts Average line-line voltages: v 12 Ts = v 10 Ts v 20 Ts = d 1 d 2 v 23 Ts = v 20 Ts v 30 Ts = d 2 d 3 v 31 Ts = v 30 Ts v 10 Ts = d 3 d 1 v Ts v Ts v Ts v 20 v 30 0 d 1 T s T s t v v 20 Ts = d 2 v Ts 0 d 2 T s T s t v 0 0 Average switch output-side currents: i 1 Ts = d 1 i a Ts i 2 Ts = d 2 i b Ts i 3 Ts = d 3 i c Ts v 30 Ts = d 3 v Ts 0 d 3 T s T s t Conducting Q devices: 1 / D 1 Q 4 / D 4 Q 2 / D 2 Q 5 / D 5 Q 3 / D 3 Q 6 / D 6 0 Fundamentals of Power Electronics 52 Chapter 17: The Ideal Rectifier

Averaged circuit model L i a Ts ø a ø b L i b Ts (d 1 d 2 ) v Ts d 1 i a Ts d 2 i b Ts d 3 i c Ts C Load v Ts ø c L (d 2 d 3 ) v Ts (d 3 d 1 ) v Ts i c Ts v 12 Ts = v 10 Ts v 20 Ts = d 1 d 2 v 23 Ts = v 20 Ts v 30 Ts = d 2 d 3 v 31 Ts = v 30 Ts v 10 Ts = d 3 d 1 v Ts v Ts v Ts i 1 Ts = d 1 i a Ts i 2 Ts = d 2 i b Ts i 3 Ts = d 3 i c Ts Q: How to vary d such that the desired ac and dc waveforms are obtained? Solution is not unique. Fundamentals of Power Electronics 53 Chapter 17: The Ideal Rectifier

Sinusoidal PWM A simple modulation scheme: Sinusoidal PWM Vary duty cycles sinusoidally, in synchronism with ac line d 1 =D 0 1 2 D m sin ωt ϕ d 2 =D 0 1 2 D m sin ωt ϕ 120 d 3 =D 0 1 2 D m sin ωt ϕ 240 where ω is the ac line frequency D 0 is a dc bias D m is the modulation index For D 0 = 0.5, D m in the above equations must be less than 1. The modulation index is defined as one-half of the peak amplitude of the fundamental component of the duty cycle modulation. In some other modulation schemes, it is possible that D m > 1. Fundamentals of Power Electronics 54 Chapter 17: The Ideal Rectifier

Solution, linear sinusoidal PWM If the switching frequency is high, then the inductors can be small and have negligible effect at the ac line frequency. The averaged switch voltage and ac line voltage are then equal: v 12 Ts = d 1 d 2 v Ts v ab Substitute expressions for duty cycle and ac line voltage variations: 1 2 D m sin ωt ϕ sin ωt ϕ 120 v Ts = V M sin ωt sin ωt 120 For small L, ϕ tends to zero. The expression then becomes 1 2 D mv = V M Solve for the output voltage: V = 2V M D m V = 2 3 V L,pk D m = 1.15 V L,pk D m Fundamentals of Power Electronics 55 Chapter 17: The Ideal Rectifier

Boost rectifier with sinusoidal PWM V = 2 3 V L,pk D m = 1.15 V L,pk D m With sinusoidal PWM, the dc output voltage must be greater than 1.15 times the peak line-line input voltage. Hence, the boost rectifier increases the voltage magnitude. Fundamentals of Power Electronics 56 Chapter 17: The Ideal Rectifier

Nonlinear modulation Triplen harmonics can be added to the duty ratio modulation, without appearing in the line-line voltages. 1 0.5 0 d 1 d 2 d 3-0.5 Overmodulation, in v 12 Ts /V which the modulation index -1 D m is increased beyond 1, also leads to undistorted line-line voltages provided that D m 1.15. The pulse width modulator saturates, but the duty ratio variations contain only triplen harmonics. V = V L,pk is obtained at D m = 1.15. Further increases in D m cause distorted ac line waveforms. 0 60 120 180 240 300 360 Fundamentals of Power Electronics 57 Chapter 17: The Ideal Rectifier ωt

Buck-type 3 acðdc rectifier 3øac input i a Q 1 Q 2 Q 3 i L L dc output ø a ø b i b D 1 D 2 D 3 C Load v i c Q 4 Q 5 Q 6 ø c Input filter D 4 D 5 D 6 Can produce controlled dc output voltages in the range 0 V V L,pk Requires two-quadrant voltage-bidirectional switches Exhibits greater active semiconductor stress than boost topology Can operate in inverter mode by reversal of output voltage polarity Fundamentals of Power Electronics 58 Chapter 17: The Ideal Rectifier

BuckÐboost topology 3øac input ø a i a Q 1 Q 2 Q 3 i L Q 7 D 7 dc output ø b i b D 1 D 2 D 3 L C Load v i c Q 4 Q 5 Q 6 ø c Input filter D 4 D 5 D 6 Fundamentals of Power Electronics 59 Chapter 17: The Ideal Rectifier

Cuk topology 3øac input dc output ø a L 1 i a Q 1 Q 2 Q D 3 1 D 2 D 3 C1 L 2 ø b L 1 i b D 7 Q7 C 2 Load v ø c L 1 i c Q 4 Q 5 Q 6 D 4 D 5 D 6 Fundamentals of Power Electronics 60 Chapter 17: The Ideal Rectifier

Use of three single-phase rectifiers 3øac input ø a ø b Dcdc converter with isolation Dcdc converter with isolation i dc output C v Each rectifier must include isolation between input and output Isolation transformers must be rated to carry the pulsating single-phase ac power p ac Outputs can be connected in series or parallel ø c Dcdc converter with isolation Because of the isolation requirement, semiconductor stresses are greater than in 3ø boost rectifier Fundamentals of Power Electronics 61 Chapter 17: The Ideal Rectifier

17.5.2 Some other approaches to three-phase rectification Low-harmonic rectification requires active semiconductor devices that are much more expensive than simple peak-detection diode rectifiers. What is the minimum active silicon required to perform the function of 3ø low-harmonic rectification? No active devices are needed: diodes and harmonic traps will do the job, but these require low-frequency reactive elements When control of the output voltage is needed, then there must be at least one active device To avoid low-frequency reactive elements, at least one highfrequency switch is needed So let s search for approaches that use just one active switch, and only high-frequency reactive elements Fundamentals of Power Electronics 62 Chapter 17: The Ideal Rectifier

The single-switch DCM boost 3 rectifier 3øac input L 1 i a D 7 dc output ø a D 1 D 2 D 3 ø b L 2 i b Q 1 C v L 3 i c ø c Input filter D 4 D 5 D 6 Inductors L 1 to L 3 operate in discontinuous conduction mode, in conjunction with diodes D 1 to D 6. Average input currents i a Ts, i b Ts, and i c Ts are approximately proportional to the instantaneous input line-neutral voltages. Transistor is operated with constant duty cycle; slow variation of the duty cycle allows control of output power. Fundamentals of Power Electronics 63 Chapter 17: The Ideal Rectifier

The single-switch DCM boost 3 rectifier 3øac input L 1 i a D 7 dc output v an V M ø a D 1 D 2 D 3 ø b L 2 L 3 i b i c Q 1 C v t ø c Input filter D 4 D 5 D 6 i a v an dt s L t Fundamentals of Power Electronics 64 Chapter 17: The Ideal Rectifier

The single-switch 3 DCM flyback rectifier 3øac input dc output ø a T 1 i a D 1 D 2 D 3 T 1 D 7 D 8 D 9 ø b T 2 i b Q 1 T 2 C v T 3 i c T 3 ø c Input filter D 4 D 5 D 6 D 10 D 11 D 12 Fundamentals of Power Electronics 65 Chapter 17: The Ideal Rectifier

The single-switch 3 DCM flyback rectifier This converter is effectively three independent single-phase DCM flyback converters that share a common switch. Since the open-loop DCM flyback converter can be modeled as a Loss- Free Resistor, three-phase low-harmonic rectification is obtained naturally. v an V M t Basic converter has a boost characteristic, but buck-boost characteristic is possible (next slide). i a v an dt s L t Inrush current limiting and isolation are obtained easily. High peak currents, needs an input EMI filter Fundamentals of Power Electronics 66 Chapter 17: The Ideal Rectifier

3 Flyback rectifier with buck-boost conversion ratio T 1 T 2 T 3 3øac input dc output ø a i a D 1 D 2 D 3 D 7 D 8 D 9 ø b i b Q 1 C v i c T 1 T 2 T 3 ø c input filter D 4 D 5 D 6 T 1 T 2 T 3 Fundamentals of Power Electronics 67 Chapter 17: The Ideal Rectifier

Single-switch three-phase zero-currentswitching quasi-resonant buck rectifier 3øac input L r i a Q 1 L dc output ø a D 1 D 2 D 3 ø b L r i b D 7 C r C v L r i c ø c Input filter D 4 D 5 D 6 Inductors L r and capacitor C r form resonant tank circuits, having resonant frequency slightly greater than the switching frequency. Turning on Q 1 initiates resonant current pulses, whose amplitudes depend on the instantaneous input line-neutral voltages. When the resonant current pulses return to zero, diodes D 1 to D 6 are reverse-biased. Transistor Q 1 can then be turned off at zero current. Fundamentals of Power Electronics 68 Chapter 17: The Ideal Rectifier

Single-switch three-phase zero-currentswitching quasi-resonant buck rectifier i a v an R 0 t Input line currents are approximately sinusoidal pulses, whose amplitudes follow the input line-neutral voltages. Lowest total active semiconductor stress of all buck-type 3ø low harmonic rectifiers Fundamentals of Power Electronics 69 Chapter 17: The Ideal Rectifier

Multiresonant single-switch zero-current switching 3 buck rectifier 3øac input i a L a Q 1 L d dc output ø a ø b i b L a C r1 v cra D 1 D 2 D 3 L r C v ø c i c L a C r1 C r1 D 4 D 5 D 6 D 7 C r2 Inductors L r and capacitors C r1 and C r2 form resonant tank circuits, having resonant frequency slightly greater than the switching frequency. Turning on Q 1 initiates resonant voltage pulses in v cra, whose amplitudes depend on the instantaneous input line-neutral currents i a to i c. All diodes switch off when their respective tank voltages reach zero. Transistor Q 1 is turned off at zero current. Fundamentals of Power Electronics 70 Chapter 17: The Ideal Rectifier

Multiresonant single-switch zero-current switching 3 buck rectifier v cra ~ i a R 0 t Input-side resonant voltages are approximately sinusoidal pulses, whose amplitudes follow the input currents. Input filter inductors operate in CCM. Higher total active semiconductor stress than previous approach, but less EMI filtering is needed. Low THD: < 4% THD can be obtained. Fundamentals of Power Electronics 71 Chapter 17: The Ideal Rectifier

Harmonic correction Nonlinear load ø a 3ø ac ø b ø c Harmonic corrector Fundamentals of Power Electronics 72 Chapter 17: The Ideal Rectifier

Harmonic correction An active filter that is controlled to cancel the harmonic currents created by a nonlinear load. Does not need to conduct the average load power. Total active semiconductor stress is high when the nonlinear load generates large harmonic currents having high THD. In the majority of applications, this approach exhibits greater total active semiconductor stress than the simple 3ø CCM boost rectifier. Fundamentals of Power Electronics 73 Chapter 17: The Ideal Rectifier

17.6 Summary of key points 1. The ideal rectifier presents an effective resistive load, the emulated resistance R e, to the ac power system. The power apparently consumed by R e is transferred to the dc output port. In a three-phase ideal rectifier, input resistor emulation is obtained in each phase. In both the single-phase and three-phase cases, the output port follows a power source characteristic, dependent on the instantaneous ac input power. Ideal rectifiers can perform the function of low-harmonic rectification, without need for low-frequency reactive elements. 2. The dc-dc boost converter, as well as other converters capable of increasing the voltage, can be adapted to the ideal rectifier application. A control system causes the input current to be proportional to the input voltage. The converter may operate in CCM, DCM, or in both modes. The mode boundary is expressed as a function of R e, 2L/T s, and the instantaneous voltage ratio v g /V. A well-designed average current controller leads to resistor emulation regardless of the operating mode; however, other schemes discussed in the next chapter may lead to distorted current waveforms when the mode boundary is crossed. Fundamentals of Power Electronics 74 Chapter 17: The Ideal Rectifier

Summary of key points 3. In a single-phase system, the instantaneous ac input power is pulsating, while the dc load power is constant. Whenever the instantaneous input and output powers are not equal, the ideal rectifier system must contain energy storage. A large capacitor is commonly employed; the voltage of this capacitor must be allowed to vary independently, as necessary to store and release energy. A slow feedback loop regulates the dc component of the capacitor voltage, to ensure that the average ac input power and dc load power are balanced. 4. RMS values of rectifiers waveforms can be computed by double integration. In the case of the boost converter, the rms transistor current can be as low as 39% of the rms ac input current, when V is close in value to V M. Other converter topologies such as the buck-boost, SEPIC, and Cuk converters exhibit significantly higher rms transistor currents but are capable of limiting the converter inrush current. Fundamentals of Power Electronics 75 Chapter 17: The Ideal Rectifier

Summary of key points 5. In the three-phase case, a boost-type rectifier based on the PWM voltagesource inverter also exhibits low rms transistor currents. This approach requires six active switching elements, and its dc output voltage must be greater than the peak input line-to-line voltage. Average current control can be used to obtain input resistor emulation. An equivalent circuit can be derived by averaging the switch waveforms. The converter operation can be understood by assuming that the switch duty cycles vary sinusoidally; expressions for the average converter waveforms can then be derived. 6. Other three-phase rectifier topologies are known, including six-switch rectifiers having buck and buck-boost characteristics. In addition, threephase low-harmonic rectifiers having a reduced number of active switches, as few as one, are discussed here. Fundamentals of Power Electronics 76 Chapter 17: The Ideal Rectifier